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Solving systems of coupled nonlinear Atangana–Baleanu-type fractional differential equations
Boundary Value Problems volume 2022, Article number: 101 (2022)
Abstract
In this work, we investigate two types of boundary value problems for a system of coupled Atangana–Baleanu-type fractional differential equations with nonlocal boundary conditions. The fractional derivatives are applied to serve as a nonlocal and nonsingular kernel. The existence and uniqueness of solutions for proposed problems using Krasnoselskii’s and Banach’s fixed-point approaches are established. Moreover, nonlinear analysis is used to build the Ulam–Hyers stability theory. Subsequently, we discuss two compelling examples to demonstrate the utility of our study.
1 Introduction
In many seemingly diverse areas of science and industry, fractional calculus and its possible applications have emerged as an effective modeling tool for many complicated phenomena [1–8]. Some academics have acknowledged the requirement of developing the idea of fractional calculus by searching for novel fractional derivatives (FDs) with various singular or nonsingular kernels to simulate a variety of actual situations in several disciplines of science and engineering. With their contributions to physical phenomena and their success in solving issues in the real world, new fractional operators have emerged from this perspective as the most effective instrument for many professionals and researchers. Before 2015, there were only singular kernels for all fractional derivatives. As a result, simulating physical phenomena with these singularities is challenging.
Caputo and Fabrizio [9] investigated a novel FD in the exponential kernel in 2015. Losada and Nieto covered some of this novel type’s characteristics in [10]. Atangana and Baleanu (AB) have looked into a novel and intriguing FD with Mittag-Leffler kernels in [11]. The kind examined by AB was extended by Abdeljawad in [12], which formed their related integral operators for orders ranging from \((0,1)\) to higher arbitrary orders. Additionally, he studied the existence and uniqueness (EaU) theorems for the Caputo-type (ABC) FD and Riemann-type (ABR), as well as some initial value problems of higher arbitrary order. Also, he established a Lyapunov-type inequality for the ABR fractional boundary value problems of order \(\rho \in (2,3]\) in the frame of Mittag-Leffler kernels. In [13, 14], Abdeljawad and Baleanu discussed the discrete forms of those novel operators. We recommend the reader to check out a number of papers [15–17] for some theoretical works on AB fractional differential equations (FDEs).
Recently, four fractional integrodifferential equations (FIDEs) were solved by Baleanu et al. [18], while the numerical solutions were presented in the works of [19–22] under suitable conditions. The dimension of the set of solutions for the second FIDE is also shown to be infinite-dimensional under a few different conditions. Aydogan et al. [23] used the Caputo–Fabrizio fractional derivative [24] to create two new high-order derivatives known as the CFD and DCF. They also looked into the possibility of solutions for two of these high-order FIDEs.
Guoa et al. [25, 26] investigated the Hyers–Ulam (HU) stability of FDEs with impulse and fractional Brownian motion under nonlocal circumstances using the semigroups of operators and the Mönch fixed-point (FP) methods. Numerous researchers examined the UH stability of solutions for particular FDEs of problems with initial or boundary stipulations in the publication series [27–32].
Recently, the ABC-type pantograph FDEs with nonlocal conditions were investigated by Abdo et al. [33] as follows:
Abdo et al. [34] investigated the ABC-type implicit FDEs as follows:
with nonlinear integral conditions
Motivated by the previously mentioned results, in this paper, we plan to investigate and create the required circumstances for the EaU and UH stability results for new systems of coupled ABR-type and ABC-type FDEs defined as follows:
where \({}^{ABR}D_{c^{+}}^{\rho }\theta (\ell )\) and \({}^{ABC}D_{c^{+}}^{\upsilon }\theta (\ell )\) represent the ABR and ABC FD of order \(\rho \in (2,3]\) and \(\upsilon \in (1,2]\), respectively, \(0<\varsigma \leq 1\) and \(\hbar :[c,d]\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) is a continuous function. To the extent that we are aware, our investigation is the first attempt to deal with high-order FDs of ABC-type and ABR-type, particularly when applied to AB integral conditions. Additionally, we looked at EaU without relying on the semigroup property. Therefore, our results will be a valuable addition to the structure of research already done on these interesting operators.
2 Basic concepts
Here, we assume that \(C(D,\mathbb{R} )\) is the space of all continuous functions \(\theta :J\rightarrow \mathbb{R} \) endowed with \(\Vert \theta \Vert =\max \{ \vert \theta ( \ell ) \vert :\ell \in D \} \), where \(D=[a,b]\). Clearly, \(( C(D,\mathbb{R} ), \Vert \cdot \Vert ) \) is a Banach space.
Definition 2.1
([11])
Assume that \(\rho \in (0,1]\). For a function θ the left-sided ABC and ABR FD of order ρ are given by
respectively, where \(\phi ( \rho ) \) refers to the normalization function so that \(\phi (0)=1=\phi (1)\) and \(\nabla _{\rho }\) represents the Mittag-Leffler function described as:
Moreover, the equivalent fractional integral of AB is given by
Lemma 2.2
([13])
Let \(\rho \in (0,1]\). If the ABC FD exists, then we get
Definition 2.3
([12])
The relation between the ABR and ABC FD is described as
Remark 2.4
If we take \({}^{AB}I_{c^{+}}^{\rho }\) on both sides of the equation (2.2) and use Lemma 2.2, we have
Lemma 2.5
([11])
For \(\theta >0\), \({}^{AB}I_{c^{+}}^{\rho }:C(D,\mathbb{R} )\rightarrow C(D,\mathbb{R} )\) is bounded.
Definition 2.6
([12])
Let \(\rho \in (k,k+1]\) and θ be such that \(\theta ^{(k)}\in H^{1}(c,d)\). Put \(\eta =\rho -k\). Then \(\eta \in (0,1]\), and we define
Further, the AB fractional integral is described as
Lemma 2.7
([12])
For \(\theta (\ell )\) defined on D and \(\rho \in (n,n+1]\), for some \(n\in N_{0}\), the assertions below are true:
-
(1)
\(( {}^{ABC}D_{c^{+}}^{\rho }{}^{AB}I_{c^{+}}^{\rho }\theta ) (\ell )=\theta (\ell )\);
-
(2)
\(( {}^{AB}I_{c^{+}}^{\rho }{}^{ABR}D_{c^{+}}^{\rho }\theta ) (\ell )=\theta (\ell )-\sum_{j=0}^{n-1} \frac{\theta ^{(j)}(c)}{j!} ( \ell -c ) ^{j}\);
-
(3)
\(( {}^{AB}I_{c^{+}}^{\rho }{}^{ABC}D_{c^{+}}^{\rho }\theta ) (\ell )=\theta (\ell )-\sum_{j=0}^{n-1} \frac{\theta ^{(j)}(c)}{j!} ( \ell -c ) ^{j}\).
Lemma 2.8
Consider \(\rho \in (n,n+1]\). Then for \(k=1,2,\ldots,n-1\), \({}^{ABR}D_{c^{+}}^{\rho } ( \ell -c ) ^{k}=0\).
Proof
Based on Definition 2.6 after putting \(\theta (\ell )= ( \ell -c ) ^{k}\), we have
As \(k< n\in \mathbb{N} \), we get \([ \frac{d}{d\ell } ] ^{(n)} ( \ell -c ) ^{k}=0\). Consequently, \({}^{ABC}D_{c^{+}}^{\rho }\theta (\ell )=0\). □
Theorem 2.9
([34])
Let Q be a closed subspace of a Banach space P and M be a mapping such that
i.e., M is a contraction mapping. Then M has an FP on Q.
Theorem 2.10
([35])
Assume that Q is a non-empty, closed, convex, and bounded subset of a Banach space P. Let the operators \(M^{1}\) and \(M^{2}\) be such that
-
for all \(\ell ,\widehat{\ell }\in P\), \(M^{1}\ell +M^{2}\widehat{\ell }\in P\);
-
\(M^{1}\) is equicontinuous;
-
\(M^{2}\) is a contraction.
Then there is a function \(z\in Q\) so that \(z=M^{1}z+M^{2}z\).
Lemma 2.11
([12])
Suppose that \(\rho \in (2,3]\) and \(\hbar \in C ( D,\mathbb{R} ) \). The problem below
has a solution
where
and
Lemma 2.12
([12])
Assume that \(\rho \in (1,2]\) and \(\hbar \in C ( D,\mathbb{R} ) \). The linear problem
has a solution
where
3 Equivalent integral equations for the proposed problem (1.1)
Lemma 3.1
Assume that \(\rho \in (2,3]\), \(\varsigma \in (0,1]\), \(\Xi =(d-c)-{}^{AB}I_{c^{+}}^{\varsigma }(\xi -c)\neq 0\) and \(\hbar \in C ( D\times \mathbb{R} ^{2},\mathbb{R} ) \). The functions θ and ϱ are a solution to the coupled ABR problem (1.1) if the coupled \((\theta ,\varrho ) \) fulfills the fractional integral equations (FIEs) below:
Proof
Let θ be a solution of the first equation of (3.1). Then by Lemma 2.11, we get
From the condition \(\theta (c)=0\), we have \(a_{1}=0\). Hence the equation (3.2) can be written as
On both sides of (3.3) applying \({}^{AB}I_{c^{+}}^{\varsigma } \), one can write
Putting \(\ell =\xi \), we have
It follows from the condition \(\theta (d)={}^{AB}I_{c^{+}}^{\varsigma }\theta (\xi )\), (3.3), and (3.4) that
Applying the value of \(a_{1}\) and \(a_{2}\) in (3.2), we obtain that
Following the same approach as the previous, one can easily obtain
On the contrary, suppose that θ fulfills (3.5). Then taking the operator \({}^{ABR}D_{c^{+}}^{\rho }\) on both sides of (3.5) and helping Lemma 2.7 and 2.8, we have
Analogously, one can write
Next, letting \(\ell \rightarrow c\) in (3.5) and (3.6), we have \(\theta ( c ) =0=\varrho ( c ) \). Conversely, taking \({}^{AB}I_{c^{+}}^{\varsigma }\) on both sides of (3.5) and (3.6), we obtain that
and
Putting \(\ell =\xi \) in (3.7) and (3.8), we have
and
From the definition of Ξ, equations (3.9) and (3.10) can be written as
And
Thus, the coupled AB fractional integral conditions are fulfilled. □
The following theorem results from Lemma 3.1:
Theorem 3.2
Suppose that \(\rho \in (2,3]\), \(\varsigma \in (0,1]\), \(\Xi =(d-c)-{}^{AB}I_{c^{+}}^{\varsigma }(\xi -c)\neq 0\) and \(\hbar :D\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) is a continuous function. Then the supposed problem (1.1) are comparable to the FIEs below:
and
where
Proof
In light of Lemma 3.1, we find that (3.1) holds. From the definitions of \({}^{AB}I_{c^{+}}^{\varsigma }\), \(\varsigma \in (0,1)\) described in (2.1) and \({}^{AB}I_{c^{+}}^{\rho }\), \(\varsigma \in (2,3]\) described in (2.3), the coupled system (3.1) can be written as
which yields that
Hence
By the same scenario, one can obtain
This completes the proof. □
For short, let us consider
4 EaU of solutions for problem (1.1)
This part is devoted to showing the EaU of solutions for the suggested problem (1.1).
Theorem 4.1
For a continuous function \(\hbar :D\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \), assume that
Then the coupled ABR-type FDEs (1.1) have a unique solution on D provided that the condition below holds
Proof
Describe an operator \(\chi :C ( D,\mathbb{R} ) \times C ( D,\mathbb{R} ) \rightarrow C ( D,\mathbb{R} ) \) as
Let us consider a closed ball \(\Lambda _{\varphi }\) so that
with radius \(\varphi \geq \frac{\mho _{1}}{1-\mho }\), where \(\mho _{1}= ( \frac{ ( d-c ) }{\Xi }\wp _{1}+\wp _{2} ) \pi _{\hbar }\) and \(\pi _{\hbar }=\sup_{\ell \in D} \vert \hbar ( \ell ,0,0 ) \vert \). Firstly, we prove that \(\chi \Lambda _{\varphi }\subset \Lambda _{\varphi }\). For each \(\theta ,\varrho \in \Lambda _{\varphi }\) and \(\ell \in D\), we get
Using (4.1), we have
Hence
Analogously, we can write \(\Vert \chi ( \varrho ,\theta ) \Vert < \varphi \). Therefore, \(\chi \Lambda _{\varphi }\subset \Lambda _{\varphi }\). Next, we shall prove that χ is a contraction. Assume that \(\varrho ,\theta ,\widehat{\varrho },\widehat{\theta }\in C ( D,\mathbb{R} ) \) and \(\ell \in D\). Then, we get
Based on our condition (4.1)
we have
Similarly, one can obtain that
Because of \(\mho <1\), we conclude that the operator χ is a contraction. Hence, Theorem 2.9 tells us there is a FP of χ in D. □
Theorem 4.2
There is at least one solution to the considered problem (1.1) in D provided that all hypotheses of Theorem 4.1are satisfied.
Proof
Divide the operator χ defined in Theorem 4.1 into two operators as
where
and
Consider a closed ball \(\Lambda _{\varphi }\), which is what Theorem 4.1 defines. In order to apply Theorem 2.10, we will break up the proof into the steps below:
-
(i)
We prove that \(\chi _{1} ( \theta ,\varrho ) +\chi _{2} ( \theta , \varrho ) \in \Lambda _{\varphi }\), for all \(\theta ,\varrho \in \Lambda _{\varphi }\). For \(\chi _{1}\in \Lambda _{\varphi }\), we get
$$\begin{aligned} & \bigl\vert \chi _{1} ( \theta ,\varrho ) ( \ell ) \bigr\vert \\ &\quad \leq \frac{ ( \ell -c ) }{\Xi } \biggl\{ \Omega _{1} \int _{c}^{\xi } ( \xi -\varpi ) \bigl\vert \hbar \bigl( \varpi ,\theta (\varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi \\ &\qquad {}+ \frac{\Omega _{2}}{\Gamma (\rho -2)} \int _{c}^{\xi } ( \xi - \varpi ) ^{\rho -1} \bigl\vert \hbar \bigl( \varpi ,\theta ( \varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi \\ &\qquad {}+\frac{\Omega _{3}}{\Gamma (\varsigma )} \int _{c}^{\xi } ( \xi -\varpi ) ^{\varsigma -1} \int _{c}^{ \varpi } ( \varpi -\varkappa ) \bigl\vert \hbar \bigl( \varkappa ,\theta (\varkappa ),\varrho (\varkappa ) \bigr) \bigr\vert \,d\varkappa \,d\varpi \\ &\qquad {}+\frac{\Omega _{4}}{\Gamma (\varsigma +\rho -2)} \int _{c}^{ \xi } ( \xi -\varpi ) ^{\varsigma +\rho -1} \bigl\vert \hbar \bigl( \varpi ,\theta (\varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi \\ &\qquad {}+\Omega _{5} \int _{c}^{d} ( d-\varpi ) \bigl\vert \hbar \bigl( \varpi ,\theta (\varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi\\ &\qquad {} + \frac{\Omega _{6}}{\Gamma (\rho -2)} \int _{c}^{d} ( d-\varpi ) ^{\rho -1} \bigl\vert \hbar \bigl( \varpi ,\theta (\varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi \biggr\} . \end{aligned}$$Applying (4.2), we have
$$ \bigl\Vert \chi _{1} ( \theta ,\varrho ) ( \ell ) \bigr\Vert \leq \frac{ ( A_{\hbar }\varphi +\pi _{\hbar } ) ( d-c ) }{\Xi }\wp _{1}. $$(4.3)Similarly
$$ \bigl\Vert \chi _{1} ( \varrho ,\theta ) \bigr\Vert \leq \frac{ ( A_{\hbar }\varphi +\pi _{\hbar } ) ( d-c ) }{\Xi } \wp _{1}. $$(4.4)Again using (4.2), we get
$$\begin{aligned} \bigl\vert \chi _{2} ( \theta ,\varrho ) ( \ell ) \bigr\vert =& \Omega _{5} \int _{c}^{\ell } ( \ell -\varpi ) \bigl\vert \hbar \bigl( \varpi ,\theta ( \varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi\\ & {} + \frac{\Omega _{6}}{\Gamma (\rho -2)} \int _{c}^{\ell } ( \ell -\varpi ) ^{\rho -1} \bigl\vert \hbar \bigl( \varpi ,\theta (\varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi \\ \leq & ( A_{\hbar }\varphi +\pi _{\hbar } ) \wp _{2}= \bigl\vert \chi _{2} ( \varrho ,\theta ) ( \ell ) \bigr\vert , \end{aligned}$$which yields that
$$ \bigl\Vert \chi _{2} ( \theta ,\varrho ) \bigr\Vert = ( A_{\hbar }\varphi +\pi _{\hbar } ) \wp _{2}= \bigl\Vert \chi _{2} ( \varrho ,\theta ) \bigr\Vert . $$(4.5)$$\begin{aligned} \bigl\Vert \chi _{1} ( \theta ,\varrho ) ( \ell ) +\chi _{2} ( \theta ,\varrho ) \bigr\Vert \leq & \bigl\Vert \chi _{1} ( \theta ,\varrho ) ( \ell ) \bigr\Vert + \bigl\Vert \chi _{2} ( \theta , \varrho ) \bigr\Vert \\ \leq & \frac{ ( A_{\hbar }\varphi +\pi _{\hbar } ) ( d-c ) }{\Xi }\wp _{1}+ ( A_{\hbar }\varphi +\pi _{\hbar } ) \wp _{2} \\ =&A_{\hbar } \biggl( \frac{ ( d-c ) }{\Xi }\wp _{1}+\wp _{2} \biggr) \varphi + \biggl( \frac{ ( d-c ) }{\Xi }\wp _{1}+ \wp _{2} \biggr) \pi _{\hbar } \\ =&\mho \varphi +\mho _{1}\pi _{\hbar }< \varphi . \end{aligned}$$By the same method one can show that \(\Vert \chi _{1} ( \varrho ,\theta ) ( \ell ) +\chi _{2} ( \varrho ,\theta ) \Vert < \varphi \). Thus, \(\chi _{1} ( \theta ,\varrho ) +\chi _{2} ( \theta , \varrho ) \in \Lambda _{\varphi }\).
-
(ii)
The mapping \(\chi _{1}\) is a contraction. The contractivity of χ finishes this step.
-
(iii)
\(\chi _{2}\) is continuous and compact. The continuity of ħ leads to the continuity of \(\chi _{2}\) and by (4.5), \(\chi _{2}\) is uniformly bounded on \(\Lambda _{\varphi }\). Now, we claim that \(\chi _{2}(\Lambda _{\varphi })\) is equicontinuous. For this, assume that \(\theta ,\varrho \in \Lambda _{\varphi }\) and \(c\leq \ell _{1}<\ell _{2}\leq d\). Then by (4.2), one can write
$$\begin{aligned} & \bigl\vert \chi _{2} ( \theta ,\varrho ) ( \ell _{2} ) -\chi _{2} ( \theta ,\varrho ) ( \ell _{1} ) \bigr\vert \\ &\quad =\Omega _{5} \biggl( \int _{c}^{\ell _{1}} \bigl[ ( \ell _{2}-\varpi ) - ( \ell _{1}-\varpi ) \bigr] \bigl\vert \hbar \bigl( \varpi , \theta (\varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi \\ &\qquad {}+ \int _{\ell _{1}}^{\ell _{2}} ( \ell _{2}-\varpi ) \bigl\vert \hbar \bigl( \varpi , \theta (\varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi \biggr) \\ &\qquad {}+\frac{\Omega _{6}}{\Gamma (\rho -2)} \int _{c}^{\ell _{1}} \bigl[ ( \ell _{2}-\varpi ) ^{\rho -1}- ( \ell _{1}- \varpi ) ^{\rho -1} \bigr] \bigl\vert \hbar \bigl( \varpi , \theta (\varpi ),\varrho (\varpi ) \bigr) \bigr\vert \,d\varpi \\ &\qquad {}+\frac{\Omega _{6}}{\Gamma (\rho -2)} \int _{\ell _{1}}^{ \ell _{2}} ( \ell _{2}-\varpi ) ^{\rho -1} \bigl\vert \hbar \bigl( \varpi ,\theta (\varpi ),\varrho ( \varpi ) \bigr) \bigr\vert \,d\varpi \\ &\quad \leq ( A_{\hbar }\varphi +\pi _{\hbar } ) \biggl( \frac{\Omega _{5} ( \ell _{2}-\ell _{1} ) }{2}+ \frac{\Omega _{6}}{\Gamma (\rho -1)} \bigl[ ( \ell _{2}-\varpi ) ^{\rho }- ( \ell _{1}-\varpi ) ^{\rho -1} \bigr] \biggr). \end{aligned}$$Hence,
$$ \bigl\Vert \chi _{2} ( \theta ,\varrho ) ( \ell _{2} ) -\chi _{2} ( \theta ,\varrho ) ( \ell _{1} ) \bigr\Vert \rightarrow 0\leftarrow \bigl\Vert \chi _{2} ( \varrho ,\theta ) ( \ell _{2} ) -\chi _{2} ( \varrho ,\theta ) ( \ell _{1} ) \bigr\Vert ,\quad\text{as }\ell _{2}\rightarrow \ell _{2}. $$
Based on the above cases and thanks to Arzela–Ascoli theorem, the relatively compactness of \(\chi _{2}(\Lambda _{\varphi })\) is verified, which proves the completely continuous of \(\chi _{2}\). From Theorem 2.10, there is at least one solution in D to the supposed problem (1.1). □
5 EaU of solutions for problem (1.2)
This part is devoted to discussing the EaU of solutions for the coupled ABC-type fractional DEs (1.2).
Lemma 5.1
Assume that \(\rho \in (1,2]\), \(\varsigma \in (0,1]\), \(\Xi =(d-c)-{}^{AB}I_{c^{+}}^{\varsigma }(\xi -c)\neq 0\) and \(\hbar _{1}\in C ( D\times \mathbb{R} ^{2},\mathbb{R} ) \). The functions θ and ϱ are a solution to the coupled ABR problem (1.2) if the coupled \((\theta ,\varrho ) \) fulfills the following FIEs:
Proof
Let θ and ϱ be the solution to the equations of (1.2). Then Lemma 2.12 and Lemma 3.1 can be used to demonstrate that the solution to equations (1.2) is given as (5.1), where \({}^{AB}I_{c^{+}}^{\rho }\hbar _{1}\) is defined in (2.4). □
Theorem 5.2
Assume that \(\rho \in (1,2]\), \(\varsigma \in (0,1]\), \(\Xi =(d-c)-{}^{AB}I_{c^{+}}^{\varsigma }(\xi -c)\neq 0\) and \(\hbar :D\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) is a continuous function. Then according to Lemma 5.1, the considered problem (1.2) is comparable to the following FIEs:
and
where
Proof
In the light of Lemma 5.1, we get (5.1). According to definitions \({}^{AB}I_{c^{+}}^{\varsigma }\) when \(\varsigma \in (0,1)\) defined in (2.1) and \({}^{AB}I_{c^{+}}^{\rho }\) when \(\rho \in (1,2]\) defined in (2.4), the coupled system (5.1) can be written as
which yields that
And
This finishes the proof. □
Theorem 5.3
Let \(\hbar :D\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function, and there is a constant number \(A_{\hbar }>0\) so that
If
then there is a unique solution to the proposed problem (1.2) in D.
Proof
Describe an operator \(\Im :C ( D,\mathbb{R} ) \times C ( D,\mathbb{R} ) \rightarrow C ( D,\mathbb{R} ) \) as
Let \(\Phi _{\gimel }\) be a closed ball defined by
with radius \(\gimel \geq \frac{\mho _{3}}{1-\mho _{2}}\), where \(\mho _{3}= ( \frac{ ( d-c ) }{\Xi }\wp _{3}+\wp _{4} ) \pi _{\hbar }\) and \(\pi _{\hbar }=\sup_{\ell \in D} \vert \hbar ( \ell ,0,0 ) \vert \). Now, we show that \(\Im \Phi _{\gimel }\subset \Phi _{\gimel }\). For each \(\theta ,\varrho \in \Phi _{\gimel }\), \(\ell \in D\), by (5.2) and helping the proof of Theorem 4.1, we have
Similarly, one can write \(\Vert \Im ( \varrho ,\theta ) \Vert < \gimel \). Hence, \(\Im \Phi _{\gimel }\subset \Phi _{\gimel }\). Next, we claim that ℑ is a contraction. Let \(\varrho ,\theta ,\widehat{\varrho },\widehat{\theta }\in C ( D,\mathbb{R} ) \) and \(\ell \in D\). Then, we have
From our assumption (5.2), one sees that
In the same approach, we can obtain
Since \(\mho _{2}<1\), then the operator ℑ is contraction. So by Theorem 2.9, we conclude that there is a unique fixed point of ℑ on D. Therefore, the unique solution for the considered problem (1.2) exists in D. □
6 Stability results
This section covers the UH and GUH stabilities for the coupled ABR-type FDEs (1.1) and ABC-type FDEs (1.2). To reach our desired goal, let us consider for \(\varepsilon >0\), the following inequalities hold
Now, we introduce the definition below:
Definition 6.1
The coupled ABR-type FDEs (1.1) are UH stable if there is a real number \(S_{\hbar }>0\) so that for each \(\varepsilon >0\) and for a solution \(\theta ,\varrho \in C(D,\mathbb{R} )\) of (6.1), there is a unique solution \(\overline{\theta },\overline{\varrho }\in C(D,\mathbb{R} )\) of the suggested problem (1.1) so that
Also, the coupled ABR-type FDEs (1.1) have GUH stable if there is a function \(\alpha _{\hbar }:\mathbb{R} ^{+}\rightarrow \mathbb{R} ^{+}\) with \(\alpha _{\hbar }(0)=0\) so that
Remark 6.2
The functions \(\overline{\theta },\overline{\varrho }\in C(D,\mathbb{R} )\) are a solution to the inequalities (6.1) if we can find a continuous functions \(U,V:D\rightarrow \mathbb{R} \) depending on θ and ϱ, respectively, such that
-
(i)
\(\vert U ( \ell ) \vert \leq \frac{\varepsilon }{2}\) and \(\vert V ( \ell ) \vert \leq \frac{\varepsilon }{2} \), for all \(\ell \in D\);
-
(ii)
\({}^{ABR}D_{c^{+}}^{\rho }\theta ( \ell ) =\hbar ( \ell ,\theta ( \ell ) ,\varrho ( \ell ) ) +U ( \ell ) \), \(\ell \in D\);
-
(iii)
\({}^{ABR}D_{c^{+}}^{\rho }\varrho ( \ell ) =\hbar ( \ell ,\varrho ( \ell ) ,\theta ( \ell ) ) +V ( \ell ) \), \(\ell \in D\).
Lemma 6.3
If θ̅ and ϱ̅ are a solution to the inequalities (6.1), and θ̅ and ϱ̅ satisfy the following inequalities:
and
where
and
Proof
In the light of Remark 6.2, we have
-
(1)
\({}^{ABR}D_{c^{+}}^{\rho }\theta ( \ell ) =\hbar ( \ell ,\theta ( \ell ) ,\varrho ( \ell ) ) +U ( \ell ) \),
-
(2)
\({}^{ABR}D_{c^{+}}^{\rho }\varrho ( \ell ) =\hbar ( \ell ,\varrho ( \ell ) ,\theta ( \ell ) ) +V ( \ell ) \),
-
(3)
\(\theta (c)=0=\varrho (c)\), \(\theta (d)={}^{AB}I_{c^{+}}^{\varsigma }\theta (\xi )\) and \(\varrho (d)={}^{AB}I_{c^{+}}^{\varsigma }\varrho (\xi )\).
Then by Theorem 3.2, we have
which implies that
Similarly, one can write
This finishes the required. □
Theorem 6.4
Let condition (5.2) hold. Then the coupled ABR-type FDEs (1.1) are UH stable provided that the assumption below is true
Proof
Assume that \(\varepsilon >0\) and \(\theta ,\varrho \in C(D,\mathbb{R} )\) are functions justifying inequalities (6.1). Let \(\overline{\theta },\overline{\varrho }\in C(D,\mathbb{R} )\) be a unique solution to the following coupled system:
Then by Theorem 3.2, we have
and
From our assumptions in (6.3), we have \(\Phi _{\theta }=\Phi _{\overline{\theta }}\) and \(\Phi _{\varrho }=\Phi _{\overline{\varrho }}\). Hence by Lemma 6.3 and (5.2), we obtain
Similarly,
Combining (6.4) and (6.5), we have
or equivalently
where
Now, if we consider \(\alpha _{\hbar }(\varepsilon )=S_{\hbar }\varepsilon \) so that \(\alpha _{\hbar }(0)=0\), then the coupled ABR-type FDEs (1.1) are GUH stable. □
Theorem 6.5
Assume that hypothesis (5.2) holds. Then the coupled ABC-type FDEs (1.2) are UH stable if the following assumption is satisfied
Proof
In the same way as for the proof of Theorem 6.4, we can write
where
Also, if we put \(\alpha _{\hbar }(\varepsilon )=S_{\hbar }^{\ast }\varepsilon \) so that \(\alpha _{\hbar }(0)=0\), then the coupled ABC-type FDEs (1.2) are GUH stable. □
7 Supportive examples
In this section, we support Theorems 4.1, 5.3, 6.4, and 6.5 through the following examples:
Example 7.1
For \(\rho \in (2,3]\), consider the following system:
Here, \(D=[0,1]\), \(\rho =\frac{5}{2}\in (2,3]\), \(c=0\), \(d=1\), \(\varsigma =\xi =\frac{1}{2}\) and
If we consider \(\ell \in [ 0,1]\) and \(\theta ,\varrho ,\overline{\theta },\overline{\varrho }\in \mathbb{R} \), we find that
Hence, condition (4.1) is satisfied with \(A_{\hbar }=\frac{1}{10}\). Also, \(\wp _{1}\approx 0.4975\), \(\wp _{2}\approx 0.4071\), \(\Xi \approx 0.9093\) and \(\mho \approx 0.09542<1\). Then all hypotheses of Theorem 4.1 are fulfilled, and hence, the coupled ABR fractional problem (7.1) has a unique solution on \([0,1]\). Further, for each \(\varepsilon =\max \{ \varepsilon _{1},\varepsilon _{2} \} \) and each \(\theta ,\varrho \in C([0,1],\mathbb{R} )\) satisfying
there are a solution \(\overline{\theta },\overline{\varrho }\in C([0,1],\mathbb{R} )\) of the coupled ABR fractional problem (7.1) with
where \(S_{\hbar }\) can be easily calculated from
Therefore, all requirements of Theorem 6.4 are satisfied. Hence, the coupled ABR fractional problem (7.1) is UH stable.
Example 7.2
Let \(\rho \in (1,2]\) and consider the following problem
Here, \(D=[0,1]\), \(\rho =\frac{3}{2}\in (2,3]\), \(c=0\), \(d=1\), \(\varsigma =\xi =\frac{1}{2}\) and
If we consider \(\ell \in [ 0,1]\) and \(\theta ,\varrho ,\overline{\theta },\overline{\varrho }\in \mathbb{R} \), we find that
Therefore, condition (5.2) is fulfilled with \(A_{\hbar }=\frac{1}{10}\). Also, \(\wp _{3}\approx 0.5104\), \(\wp _{4}\approx 0.2618\), \(\Xi \approx 0.82798\) and \(\mho _{2}\approx 0.08782<1\). Hence, all assumptions of Theorem 5.3 are fulfilled. Thus, the coupled ABC fractional problem (7.2) has a unique solution on \([0,1]\). In addition, for each \(\varepsilon =\max \{ \varepsilon _{1},\varepsilon _{2} \} \) and each \(\theta ,\varrho \in C([0,1],\mathbb{R} )\) satisfying
there are a solution \(\overline{\theta },\overline{\varrho } \in C([0,1],\mathbb{R} )\) of the coupled ABC fractional problem (7.2) with
where \(S_{\hbar }\) can be easily calculated from
Therefore, all requirements of Theorem 6.5 are satisfied. Hence, the coupled ABC fractional problem (7.2) is UH stable.
8 Conclusion and future works
Researchers who have examined and established some qualitative aspects of solutions to FDEs utilizing such operators have recently become interested in the theory of fractional operators in the framework of Atangana and Baleanu.
The EaU of solutions to the coupled nonlinear AB FDEs subject to integral stipulations have been devised and studied to serve this goal. Additionally, stability analysis from mathematical analytical methods has been examined. To support our findings, two examples are provided. Our method in this study is novel because it relies on a minimal number of hypotheses and allows us to provide the results on existence, uniqueness, and UH stability without using the semigroup characteristic. Our strategy is founded on the transformation of the provided problem into an FIE and the application of several common FP theorems resulting from Banach and Krasnoselskii-type theorems. We further examined the stability findings in the UH sense using mathematical analysis approaches. In future works, we will investigate brand-new numerical discoveries relating to this operator with a higher order. We also look forward to what will happen under the Mittag-Leffler kernels.
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Abbreviations
- FDs:
-
Fractional derivatives
- AB:
-
Atangana and Baleanu
- EaU:
-
Existence and uniqueness
- ABR:
-
Atangana–Baleanu–Riemann
- ABC:
-
Atangana–Baleanu–Caputo
- FDEs:
-
Fractional differential equations
- FIDEs:
-
Fractional integro-differential equations
- CFD:
-
Caputo–Fabrizio derivative
- FP:
-
Fixed point
- HU:
-
Hyers–Ulam
- GHU:
-
Generalized Hyers–Ulam
- FIEs:
-
Fractional integral equations
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This work is funded through research groups program under grant R.G.P.2/207/43 provided by the Deanship of Scientific Research at King Khalid University, Saudi Arabia.
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Hammad, H.A., Zayed, M. Solving systems of coupled nonlinear Atangana–Baleanu-type fractional differential equations. Bound Value Probl 2022, 101 (2022). https://doi.org/10.1186/s13661-022-01684-0
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DOI: https://doi.org/10.1186/s13661-022-01684-0